where \(\psi :\mathbb {R}^+\rightarrow \mathbb {R}\)  is a function, is, up to a uniform translation in the \(x\) -variable, in the form of (REF ) with \((x,v)\in G=T\mathbb {R}^d=\mathbb {R}^d\times \mathbb {R}^d\) , with the interaction function \(\phi \)  now failing to have only eigenvalues with positive real parts, and fails to be smooth if \(\psi \)  possesses a singularity at 0. Another example is the Swarmalator model proposed by O'Keeffe, Hong, and Strogatz [1]}:
\({\left\lbrace \begin{array}{ll}\dot{x}_i = v_i+\frac{1}{N} \sum _{j=1}^N [ I_{att}(x_j-x_i) F(\theta _j-\theta _i)-I_{rep}(x_j-x_i)],\\\dot{\theta }_i = \omega _i+\frac{\kappa }{N}\sum _{j=1}^N H_{att}(\theta _j-\theta _i)G(x_j-x_i),\end{array}\right.}t>0,~i=1,\cdots ,N.\) 
